Abstract

The standard map, paradigmatic conservative system in the (x,p) phase space,
has been recently shown [Tirnakli and Borges (2016)] to exhibit interesting
statistical behaviors directly related to the value of the standard map
external parameter K.
A detailed numerical description is achieved in the present paper.
More precisely, for large values of K (e.g., K=10), the Lyapunov exponents
are neatly positive over virtually the entire phase space, and,
consistently with Boltzmann-Gibbs (BG) statistics, we verify
qent=qsen=qstat=qrel=1,
where qent is the q-index for which the nonadditive entropy
Sq≡k1−∑Wi=1pqiq−1
(with S1=SBG≡−k∑Wi=1pilnpi)
grows linearly with time before achieving its W-dependent saturation value;
qsen characterizes the time increase of the sensitivity ξ
to the initial conditions, i.e.,
ξ∼eλqsentqsen(λq%
sen>0),
where ezq≡[1+(1−q)z]1/(1−q); qstat
is the index associated with the qstat-Gaussian distribution
of the time average of successive iterations of the x-coordinate;
finally, qrel characterizes the qrel-exponential
relaxation with time of the entropy Sqent towards
its saturation value.
In remarkable contrast, for small values of K (e.g., K=0.2),
the Lyapunov exponents are virtually zero over the entire phase space,
and, consistently with q-statistics, we verify
qent=qsen=0,
qstat≃1.935,
and qrel≃1.4.
The situation corresponding to intermediate values of K,
where both stable orbits and a chaotic sea are present, is discussed as well.
The present results transparently illustrate when BG behavior
or q-statistical behavior are observed.

pacs:

Statistical characterization of dynamical systems,
particularly weakly chaotic low-dimensional dissipative maps,
have been made according to
its sensitivity to the initial conditions,
the rate of increase of the entropy,
the rate of relaxation to the attractor,
and the distribution of a statistical variable
Tsallis2009 ().
For strongly chaotic systems, these behaviors present an exponential nature
and are described by the Boltzmann-Gibbs (BG) statistical mechanical framework
BeckSchlogl ().
Weakly chaotic systems, that present zero largest Lyapunov exponent (LLE)
λ,
need a more general formalism to be properly handled.
It has been observed that the nonextensive statistical mechanical framework
is able to describe different features of various classes of complex systems
Rapisarda-2004 (); Esquivel-2010 (); CMS-2010 (); Shao-2010 (); Sotolongo-2010 (); Mohanalin-2010 (); ALICE-2010 (); PHENIXbis (); PHENIX-2011 (); ALICE-2011 (); ATLAS-2011 (); Wong-2012 (); Marques-2013 (); Bogachev-2014 (); Combe-2015 (),
including the logistic map Baldovin-Robledo-2002 ().
If λ approaches zero,
the sensitivity to the initial conditions
and the rate of relaxation to the attractor
follow q-exponential behaviors,
the distributions follow a q-Gaussian form,
and the rate of increase of entropy demands Sq entropy,
instead of BG entropy.
All these behaviors are parameterized by the indices
qsen,
qent,
qrel,
qstat,
for sensitivity,
entropy production,
relaxation,
and
stationary distribution,
respectively.
It has been observed (and in some instances proved) that,
for low-dimensional systems, qsen=qent,
through the q-generalized Pesin equality Tsallis2009 ().
Consequently, three (possibly related) quantities remain to be addressed,
that are currently referred to as the q-triplet Tsallis2009 ().
The q-triplet has been experimentally observed in a variety of complex
systems like in the ozone layer Ferri2010 () and in the
Voyager I data for the solar wind Burlaga2013 (), among many others.
Relations are expected to connect
qent, qstat, and qrel,
ultimately completely determined
by the nonlinear dynamics of the system.
The possible relations between the q-triplet indices
should in principle exhibit the fixed point q=1 (BG statistics).
At least for low-dimensional systems, if one of them equals unity,
the others also equal unity. This typically happens whenever λ>0.
In fact, LLE plays a central role in the definition of which statistical
framework is to be used in the description of nonlinear dynamical systems.

In the present work we address a low-dimensional conservative system,
namely the standard map chirikov (); zaslavsky1 ():

pi+1=pi−Ksinxi;xi+1=xi+pi+1

(1)

(p and x are taken as modulo 2π).
Its phase space can exhibit strongly chaotic regions,
characterized by positive local LLE λ,
and weakly chaotic regions, with zero values of λ.
The LLE has been numerically evaluated through
the Benettin et al algorithm benettin ().
It is calculated for each initial condition separately,
as illustrated by Fig. 2 of tirnakli-borges-2016 ()
(see also BarangerLatoraRapisarda2002 ()).
The extent of the phase space for each case (strongly or weakly chaotic)
is determined by the control parameter K.
For sufficiently small values of K, for instance K=0.2,
the entire phase space is practically stable (zero LLE),
and if K takes high values (K=10), the points of the phase space
have finite positive values of λ.
In the present work, we evaluate the values of qsen,
qent, and qrel
for both strongly and weakly chaotic cases.
We also revisit the stationary distributions, particularly the mixed case
(strongly and weakly chaotic),
that has been recently found for this system tirnakli-borges-2016 ().
Finally, we have determined the relation of strongly and weakly chaotic
qstat indices, with respect to the stationary distributions
of the map for arbitrary values of K.

where Δx(t) is the temporal dependence of the discrepancy
of two very close initial conditions at time t.
When ergodic behavior dominates over the whole phase space,
Eq. (2) is expressed by the exponential dependence,

ξ(t)=eλt

(3)

and λ is the standard maximum Lyapunov exponent.
Strongly chaotic systems are those which present λ>0.
They are strongly sensitive to initial conditions.
Insensitivity to initial conditions are described by λ<0.
The marginal case λ=0 is governed by a more subtle rule
than
Eq. (3)
tsallis-plastino-zheng-1997 ():

ξ(t)=expqsen(λqsent)

(4)

where expqx≡[1+(1−q)x]1/(1−q)+,
[u]+=max{u,0}, is the so called q–exponential function
tsallis:quimica_nova (),
and λqsen is a generalized Lyapunov exponent.
If λqsen>0 (with qsen<1)
the system is weakly chaotic, in the sense that
ξ(t)
diverges slower than the exponential (strongly chaotic) case,
— asymptotically, a power law.
If the system is strongly chaotic,
qsen→1,
and Eq. (4) recovers the standard exponential dependence,
with λqsen→λ1≡λ.

The correctness of this description can be checked, for both chaotic regimes
of standard map, in Figures 1 and 4.
They show, for different values of q, the averages of lnqξ(t)
(where lnqx≡(x1−q−1)/(1−q) is the inverse function of the
q-exponential, and ln1x=lnx)
over Nr realizations. Each realization starts with a randomly chosen
pair of very close initial conditions, that are localized inside a
particular cell of the W equally partitioned phase space.
In Fig. 1, that corresponds to the case of strong chaos
K=10, where LLE >0), we consider decreasing initial discrepancies in
Eq. (2) so as to obtain a well defined behavior for
increasingly long times.
In all cases (strongly and weakly chaotic chaos), after a transient,
we verify a nontrivial property ananos-tsallis-2004 ()
namely that exists a special value of q, noted qavsen
(where av stands for average),
which yields a linear dependence of
⟨lnqsenξ⟩
with time.
In other words, we verify
⟨lnqξ(t)⟩≈λqavsent.

For the strongly chaotic case, exponential
sensitivity to the initial conditions is verified with
qavsen=1, and the
same generalized Lyapunov exponent characterizes the whole phase space no mater
the value of W or the particular cell to be chosen. Fig. 2
exemplifies this fact, as it shows that the slope of the temporal
evolution of the averaged sensitivity to initial conditions is preserved over an
intermediate regime where correlation coefficient equals one and
λ=λqavsen=1=1.62 (time steps)−1,
in accordance with latora-baranger1999 ().
Consequently, generalized Lyapunov exponent does not depend on the spreading
occupation index latora-baranger-tsallis-rapisarda-2000 () of averaging
cells, and it makes sense choosing the random pairs of initial conditions
to compute the Nr realizations all over the phase space, i.e. W=1.

Figure 2:
(Color online)
Left: Color map of the integrated number of occupied cells
in a W=500×500 equally partitioned phase space,
from iteration 1 to iteration 10, as a function of the
(xth,pth) localization of the cell that
contains initial conditions.
Right: Temporal evolution of ⟨lnq=1ξ⟩
for K=10, where Δr≡∥Δx(0)∥=10−12
is the Euclidean distance of the Nr pairs of initial conditions.
The Nr=106 realizations are estimated over one of
the quickest spreading cells
(the cell at line 248 and column 27, green squares)
and over one of the slowest spreading cells
(the cell at line 384 and column 405, black squares),
amongst the W=500×500 cells of an equally partitioned phase
space. Averaged temporal evolution ⟨lnq=1ξ(t)⟩ over the
not partitioned phase space is also shown (W=1, red squares).

For the weakly chaotic case,
stability islands dominate the phase space.
Consequently, the characterization of the spreading occupation index of cells
is much more intricate.
In this case, a “global characterization” of the map has tentatively been made
Baldovin-2002 (); Baldovin-2004 ().
But Fig. 3 (K=0.2, where LLE = 0) demonstrates that a
common transient time can not be defined, and this fact tends to preclude a neat
linear dependence of ⟨lnqsenξ⟩ with time.
We now do not choose the Nr random pairs of initial conditions
all over the phase space, but in particular cells, i.e., in local regions.
Nevertheless, we identify regions that exhibit qavsen=0,
whose value of λqavsen depend on the
initial cell, as shown in Figure 4.
In fact, a huge set of randomly chosen cells exhibit analogous results of
qavsen=0,
but we have also found cells where linear behavior is satisfied for
qavsen≠0 (Fig. 5).

Figure 3:
(Color online)
Color maps of the integrated number of occupied cells in a W=200×200
equally partitioned phase space, from iteration 1 to iteration
t (t=7, left, t=200, right), as a function of the
(xth,pth) localization of the cell that contains
the initial conditions.
A set of quasi-periodic trajectories are superimposed to the maps,
in order to warn about the different local transient
times (yellow, black and green lines).

Figure 4:
(Color online) Temporal evolution of ⟨lnξq⟩,
when K=0.2, for (64th,64th) (left figure)
and (17th,17th) (right figure)
initial cells in the W=200×200 equally partitioned phase space.
Averages are estimated for a set of Nr=104 realizations,
whose initial conditions have been randomly chosen in their respective
initial cells.
The (17th,17th) case presents a slightly periodically
modulated evolution that is due to the not connected character of trajectories
(see Fig. 3).
Insets represent the nonlinearity measure R≡C/B in the fitting curve
of ⟨lnξq(t)⟩, A+Bt+Ct2,
and demonstrate that qavsen=0 is the optimum value
of q (a straight line fitting curve is obtained).Figure 5:
(Color online) Nonlinearity measure of the polynomial fitting curve
⟨lnξq(t)⟩ vs. q, for averages over
(17th,17th), (64th,64th)
and (72th,18th) cells of the W=200×200
equally partitioned phase space of the standard map with K=0.2.
Red and blue lines are superimposed.

With respect to the entropy production per unit time,
we may conveniently use the q-entropy (k=1, henceforth)
Tsallis1988 (); Tsallis2009 ()

Sq(t)=k1−∑Wi=1pqiq−1(S1=SBG=−kW∑i=1pilnpi).

(5)

The q-entropy production is estimated dividing the phase space in W equal
cells, and randomly choosing Nic≫W initial conditions inside one
of the W cells (typically Nic=10W).
We follow the spreading of points within the phase space, and
calculate Eq. (5) from the set of occupancy probabilities
{pi(t)} (i=1,2,…,W).
Fluctuations are of course present, but they can be reduced
when we choose Nc≫1 initial cells
within which the Nic initial conditions are chosen,
and average Sq(t) over the Nc realizations.
The proper value of the entropic parameter qavent
is the special value of q which makes the averaged q-entropy
production per unit time to be finite,
which is ultimately related with the extensivity of the entropy.
The q-entropy production per unit time is consequently obtained as

Kqavent=limt→∞limW→∞limNic→∞⟨Sqavent(t)⟩Nct<∞,

(6)

and must be calculated taking into account that the partitions of phase space,
Nc and Nic are such as to obtain robust results.

When the LLE is largely positive
and the phase space is dominated by a chaotic sea,
the Boltzmann-Gibbs entropy (S1=SBG≡−∑Wi=1pilnpi)
is expected to be the appropriate one (i.e. qent=1).
Our numerical results show, as expected, that the proper value
of the entropic index is
qavent=qavsen=1 for K=10
(see Figure 6). Both indexes have been obtained comparing
the nonlinearity measure of the polynomial fitting curves
over the intermediate regime (after a transient and before saturation)
were the correlation coefficient is constant and equals 1.
Another interesting result that emerged is the coincidence of the slopes of
the sensitivity and entropy functions of time
(see Fig. 2 and Fig. 6),
i.e., the q-entropy production per unit time satisfies
Kqavsen=1=λqavsen=1.
These results are numerically compatible with a q-generalization of a
Pesin-like identity for ensemble averages in strongly chaotic conservative maps,
and reinforce those in ananos-tsallis-2004 ().

Figure 6:
(Color online) Temporal dependence of (qavent=1)-entropy
in the standard map with K=10,
averaging over 1%W, for different values
of the number of cells in the equally partitioned phase space, W.
The case of q<qavent=1
(q>qavent=1)
corresponds to positive (negative) concavity.
The t→∞ limiting value is ⟨S1⟩=lnW.

Entropic characterization of weak chaos regime in conservative maps
is much more subtle than in dissipative maps
(see Casati-Tsallis-Baldovin-2005 (); RuizTsallis ()).
First, let us notice that one of the practical difficulties to identify
the weak chaos regime are the huge computer times required, because of the
slow convergence of the finite time Larger Lyapunov Exponent
to its t→∞ limit value.
Furthermore, we know that the phase space of weakly chaotic conservative maps
is dominated by stability islands and regions with a huge diversity
of spreading occupation indexes,
and this makes a hard task to reduce fluctuations by averaging the entropy.
We have already pointed out that spreading occupation index of cells,
and consequently their transient time scales, are essential
in the q-sensitivity characterization of the map.
Fig. 7 reveals the incidence of this phenomena on the
entropy production, which is related to the sensibility to initial conditions,
and we conclude that a smart averaging criterion must be adopted.

Initially we have tested the adequacy of the Smaller Alignment Index
(SALI) Skokos () to identify weak chaos in the K=0.2 standard map,
so as to reduce the computer times required for LLE computation.
We observe that the SALI temporal evolution is a computationally cheap tool
to detect chaotic behavior of trajectories, as it tends to zero in both ordered
and chaotic orbits, but with completely different time rates:
SALI decreases abruptly in strongly chaotic orbits and reaches the limit of
accuracy of the computer after about 200 iterations and, in contrast,
it decreases with time as a power law in ordered orbits.
We have verified, for the first time to the best of our knowledge,
that the SALI decreases with time as a power law
even in the case of weakly chaotic orbits.
Fig. 8 shows, for the standard map with K=0.2,
that the time required to characterize weak chaos of stability island
is generically shorter if we study the SALI temporal evolution instead of
the LLE t→∞ limit computation.

Figure 8:
(Color online) Left: Paradigmatic stability island in phase space
for the standard map with K=0.2,
whose respective initial conditions are specified
in the label.
Center: Log-log representation of the respective finite time
Larger Lyapunov Exponent.
Right: Log-log representation of the respective SALI temporal evolution.
The quasi-periodicity of trajectories are detected both by the
t→∞ limit of LLE, and by the power-law behavior
of the SALI temporal evolution.

We then obtain the temporal production of the averaged entropy
⟨Sq⟩Nc, selecting the Nc cells between those
that present analogous SALI temporal evolution:
the smoother the log-log SALI evolution is, the smother its entropy production.
This criterion
provides averaged q-entropies, ⟨Sq⟩Nc,
that do not present the typical oscillations observed in conservative maps
RuizTsallis ().
Fig. 9 exemplifies this fact.

Figure 9:
(Color online) Increase of (qavent=0)-entropy
in the standard map with K=0.2,
for different values of W, all over a longitudinal band of phase space
that covers p=2 (x∈[0,2π]).
The inset shows that q<qent=0 (q>qent=0)
corresponds to positive (negative) concavity.

Summarizing, for weakly chaotic dissipative systems,
particularly for the logistic map,
it has been observed that there is a specific value of q<1
(for the logistic map, qent=0.2445latora-baranger-tsallis-rapisarda-2000 ())
which yields finite entropy production per unit time,
and hence Sqent (qent<1)
is the proper one to thermodynamically characterize the system.
Verification of this behavior for a conservative system (the standard map)
is presently done.
Our numerical results show in both strongly and weakly chaotic regime,
as expected, qavent=qavsen,
and this points towards the conjecture of the generalized Pesin equality
tsallis-plastino-zheng-1997 ().
In particular, qavsen=qavent=1
and Kqavent=λqavsen=1.62
for K=10 (strong chaotic regime, as λ>0),
and qavsen=qavent=0 for K=0.2
(weakly chaotic regime,
as λ=0, λqavsen>0).
With respect to the weakly chaotic regime, a local characterization
of generalized Pesin-like remains to be made,
as Kqavent and λqavsen
have both shown to be local properties.

The methods described in
Sections II and III
take an ensemble of initial conditions
inside one of W cells of the phase space, and
thus Sq(0)=0, ∀q.
In Ref. moura-tirnakli-lyra-2000 (), the authors have
considered an ensemble of initial conditions spread over the entire
phase space of a dissipative system (particularly, the logistic map),
therefore, with the maximum possible value for the entropy
Sqent(0)=lnqentW.
The time evolution brings the system to its attractor,
and the number of occupied cells falls according to an exponential
relation, in the case of a strongly chaotic system,
or according to a q-exponential, in the case of a weakly
chaotic system.
Later it was found a relation between qrel and
qent by analyzing the rate in which the entropy
goes to its final value,
beginning with Sqent(0)=0,
instead of
Sqent(0)=lnqentWananos-borges-tsallis-oliveira-2002 ().

The method adopted in moura-tirnakli-lyra-2000 ()
cannot be identically applied to a conservative map,
since there is no attractor,
but that one introduced in ananos-borges-tsallis-oliveira-2002 ()
can be adapted to area-preserving models.
We consider the variable
ΔSq(t)=Sq(∞)−Sq(t)
and evaluate its average over an ensemble of initial conditions
for t→∞.
The average must be computed with a different criterion in strong and weak chaos
regimes, according to the respective stability distributions on the phase space.

The strongly chaotic map is dominated all over the partitioned phase space
by a chaotic sea, and we can analytically calculate
Sq=1(∞)=lnW.
In that case, the average
⟨ΔSq(t)⟩Nic has been estimated
over the most quickly spreading cells Nic
to minimize the time computation.
On the contrary, the stability islands dominate the phase space in the weakly
chaotic regime, and their respective value of
Sqent(∞)=lnqentWislands
is unknown, as it numerically depends on the cells where the initial conditions
are randomly chosen.
We must find a proper criterion for the numerical estimation
of Sqent(∞) and
⟨ΔSq(t)⟩Wislands,
and we privilege those cells that cover a region were SALI behaves
in a smoothly manner, such as to produce entropy in a smoothly manner too.

It is expected that
limW→∞ΔSqavent∼expqavrel(−t/τq),
where τq is a relaxation time.
Figures 10 and 11
show the time dependence of
lnqrel⟨ΔSqent(t)⟩/⟨Sqent(∞)⟩
for the strongly chaotic case (K=10),
and the weakly chaotic case (K=0.2), respectively.
The former case displays an exponential relaxation,
i.e., qrel=1, consistently with BG framework.
The weakly chaotic case presents a q-exponential relaxation regime,
with qrel=1.4 for K=0.2.

Figure 10:
(Color online) Left: Time evolution of
lnqrel⟨ΔSq=1(t)⟩/⟨Sq=1(∞)⟩
in the (strongly chaotic) standard map with K=10.
The averages have been calculated over the more quickly spreading cells,
to optimize the computational cost.
Right: The numerical results of the inverse of relaxation time
estimations, τ−1rel(W−1/2),
to the Sqent=1(∞) entropy limit, are compatible with a
finite relaxation time τq=1≡τrel(W→∞).
Figure 11:
(Color online)
Time evolution of
lnqrel⟨ΔSq=0(t)⟩/⟨Sq=0(∞)⟩
in the K=0.2 (weakly chaotic) standard map.
The averages are calculated over the cells that belong to the same
longitudinal band in phase space analyzed in Fig. 9.
According to a procedure similar to what was done
in Fig. 4 and 5,
the nonlinearity measure identifies qrel=1.4.

and where q→1 recovers the Gaussian distribution P1(x;μ1,σ1)=1σ1√2πe−12(x−μ1σ1)2.

Both Gaussian and q-Gaussian distributions have recently been observed
for the standard map tirnakli-borges-2016 ().
We have analyzed the distribution of sums of iterates of the map,
for representative K values, as defined by

y=T∑i=1(xi−⟨x⟩),

(10)

where the average ⟨⋯⟩ is both a time average
over T iterations and an ensemble average over M initial conditions:

⟨x⟩=1M1TM∑j=1T∑i=1x(j)i.

(11)

The probability distributions of y, namely P(y),
that emerge from the standard map with weakly chaotic phase space
are q-Gaussians, while Gaussian distributions are associated with
the strongly chaotic regime.
The scenario for intermediate values of K is also quite rich.
For these cases, the phase space displays regions with points that have
positive LLE, — the chaotic sea —,
and regions with λ=0,— the stability islands.
The probability distributions of y for initial conditions
taken inside the strongly chaotic regions are Gaussians,
and those distributions for initial conditions that belong
to the weakly chaotic regions are q-Gaussians.
One remarkable feature is that qstat=1.935 in all cases,
regardless the value of K.

If the initial conditions are spread over the entire phase space,
thus the initial conditions are from both the chaotic sea
and the stability islands,
the distribution that emerges is a linear combination of a Gaussian and
a q-Gaussian.
Consequently, the probability distribution of the standard map,
for any arbitrary value of K, can be modeled as

P(y)=αPq(y;μq,σq)+(1−α)P1(y;μ1,σ1).

(12)

where Pq(y) and P1(y) are the probability densities of y
for the initial conditions taken inside the weak and strong chaos regions,
respectively.

In Ref. tirnakli-borges-2016 ()
it was taken the linear combination of the probability distributions
normalized to the maximum value P(0),
and the physical meaning of the interpolating parameter was not clear.
Now we have taken the linear combination of the usual probability densities in
Eq. (12).
We find out the ratio of the areas for the strongly and weakly chaotic regions.
In order to achieve this, we randomly generate
Nic=4×106 initial conditions
all over the phase space and count how many of those having non-zero Lyapunov
and those having nearly zero Lyapunov.
Dividing these numbers by the total number of initial conditions,
we mimic the ratio of the areas.
We classify an initial condition belonging to a weakly chaotic region
if its Lyapunov exponent is smaller than a given threshold
λ<λ−=5×10−5 (time steps)−1.
Correspondingly, an initial condition is considered to belong to
a strongly chaotic region whenever
λ>λ+=10−2 (time steps)−1.
These limiting values λ− and λ+
are chosen according to what is explained in tirnakli-borges-2016 ():
the phase-space ratio, i.e.,
[number of points with λ<λthreshold] /
[number of points with λ>λthreshold]
remains almost constant for
λ−<λthreshold<λ+.
We call
α1=Nic(λ>λ+)/Nic, and
αq=Nic(λ<λ−)/Nic.
There remains a small amount of initial conditions
with intermediate Lyapunov exponent values
(λ−≤λ≤λ+)
that are not taken into account in Eq. (12),
and that are possibly negligible in the macroscopic limit.
We have thus considered the approximated form of
Eq. (12) as

P(y)≈αqPq(y;μq,σq)+α1P1(y;μ1,σ1).

(13)

See the results of some representative K values in the Table.

A1=0.5642..., Aq=1.935=0.3364...

K

B1

Bq

α1

αq

0.2

−

0.015

0

1

0.6

10.7×10−7

0.015

0.0380

0.9416

1

8.00×10−7

0.015

0.4826

0.5078

2

1.06×10−7

0.010

0.7622

0.2367

3

0.40×10−7

0.010

0.8825

0.1170

10

0.52×10−7

−

1

0

Table 1:
The values of the parameters for representative K values.

Then, we compute the B1 and Bq values using the pdf of the system
coming from the initial conditions with positive and with zero Lyapunov values,
respectively. After determining these values now we have no fitting parameter
and α values are entirely determined by the intrinsic dynamics
of the map.
For some representative values of K, the results are given
in Fig. 12.

The q-triplet (qent, qrel, qstat)
is an important feature in the statistical description of dynamical systems
that might be nonergodic and nonmixing, as a consequence of zero LLE.
These indices define
the rate of entropy production,
the rate of relaxation,
and the distribution of the stationary states.
Positive LLE leads to ergodic, mixing systems, and strong chaos,
and these indices collapse into q=1, within the BG framework.
Zero LLE may lead to breakdown of ergodicity,
nonmixing dynamics, and weak chaos.
The statistical description in many such circumstances appears to be associated with
the q-entropy, where the usual exponential and Gaussian functions turn into
more general forms, namely the q-exponential and the q-Gaussian ones.
This scenario has been previously observed and evaluated for dissipative
low-dimensional systems.

We have now considered a paradigmatic low-dimensional conservative system,
the standard map.
We have evaluated the q-triplet, and have verified that the sensitivity
to initial conditions index satisfies qsen=qent,
in both strongly and weakly chaotic regimes.
Our results corroborate the expected situation, i.e.,
that if the phase space is completely strongly chaotic (LLE>0),
qent=qrel=qstat=1 and
Kqavent=λqavsen=1=1.62>0.
These results are numerically compatible with a q-generalization
of a Pesin-like identity for ensemble averages,
extending those in ananos-tsallis-2004 () to the case of conservative maps.
If the entire phase space is weakly chaotic regime (λ≈0),
which happens for low values of K, the standard map is characterized by
qent=0,
qrel≃1.4, and
qstat≃1.935.
We have shown that λqavsen
is a property that can be locally characterized by the finite temporal
evolution of LLE or, alternatively, by the SALI, which has demonstrated to be
a computationally cheaper tool to identify different chaotic regimes.
The phase space for intermediate values of K features regions of
positive and regions of zero LLE.
Ensemble averages of initial conditions taken within
the strongly chaotic regions behave according to
qent=qrel=qstat=1.
In contrast, ensemble averages of initial conditions taken within
the weakly chaotic regions behave according to the q-triplet
and, at least in what concerns the stationary distribution,
with the same values of qstat.
This work extends what has been done in Ref. tirnakli-borges-2016 ().
We have numerically found the proper superposition of distributions
for the mixed case (coexistence of strongly and weakly chaotic regimes)
without any additional fitting parameter.

The q-triplet structure plays a central role
on the statistical description of dynamical systems,
and additional works addressing other conservative and dissipative maps
shall bring further tests for the robustness of this framework,
and they are very welcome.

Acknowledgments

This work has been partially supported by CNPq and Faperj
(Brazilian Agencies), and by TUBITAK (Turkish Agency)
under the Research Project number 115F492.
One of us (CT) also acknowledges partial financial support by
the John Templeton Foundation (USA).